This paper is a conceptual study in the philosophy of logic. The question considered is 'How may formulae of the propositionalcalculus be brought into a representational relation to the world?'. Four approaches are distinguished: (1) the denotational approach, (2) the abbreviational approach, (3) the truth-conditional approach, and (4) the modelling approach. (2) and (3) are very familiar, so I do not discuss them. (1), which is now largely obsolete, led to some interesting twists and turns in early (...) analytic philosophy which will come as news to many contemporary readers, so I discuss it in some detail. The modelling approach is, to the best of my knowledge, newly introduced here. I am not presenting it as a rival to the other approaches, but as a philosophically interesting possibility. (shrink)
The problem of approximating a propositionalcalculus is to find many-valued logics which are sound for the calculus (i.e., all theorems of the calculus are tautologies) with as few tautologies as possible. This has potential applications for representing (computationally complex) logics used in AI by (computationally easy) many-valued logics. It is investigated how far this method can be carried using (1) one or (2) an infinite sequence of many-valued logics. It is shown that the optimal candidate (...) matrices for (1) can be computed from the calculus. (shrink)
In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic, an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a (...)calculus for the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof. (shrink)
I am presenting a sequent calculus that extends a nonmonotonic consequence relation over an atomic language to a logically complex language. The system is in line with two guiding philosophical ideas: (i) logical inferentialism and (ii) logical expressivism. The extension defined by the sequent rules is conservative. The conditional tracks the consequence relation and negation tracks incoherence. Besides the ordinary propositional connectives, the sequent calculus introduces a new kind of modal operator that marks implications that hold monotonically. (...) Transitivity fails, but for good reasons. Intuitionism and classical logic can easily be recovered from the system. (shrink)
Propositional logics in general, considered as a set of sentences, can be undecidable even if they have “nice” representations, e.g., are given by a calculus. Even decidable propositional logics can be computationally complex (e.g., already intuitionistic logic is PSPACE-complete). On the other hand, finite-valued logics are computationally relatively simple—at worst NP. Moreover, finite-valued semantics are simple, and general methods for theorem proving exist. This raises the question to what extent and under what circumstances propositional logics represented (...) in various ways can be approximated by finite-valued logics. It is shown that the minimal m-valued logic for which a given calculus is strongly sound can be calculated. It is also investigated under which conditions propositional logics can be characterized as the intersection of (effectively given) sequences of finite-valued logics. (shrink)
By pure calculus of names we mean a quantifier-free theory, based on the classical propositionalcalculus, which defines predicates known from Aristotle’s syllogistic and Leśniewski’s Ontology. For a large fragment of the theory decision procedures, defined by a combination of simple syntactic operations and models in two-membered domains, can be used. We compare the system which employs `ε’ as the only specific term with the system enriched with functors of Syllogistic. In the former, we do not need (...) an empty name in the model, so we are able to construct a 3-valued matrix, while for the latter, for which an empty name is necessary, the respective matrices are 4-valued. (shrink)
A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of this paper is (...) to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositionalcalculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete. (shrink)
There is a long tradition in formal epistemology and in the psychology of reasoning to investigate indicative conditionals. In psychology, the propositionalcalculus was taken for granted to be the normative standard of reference. Experimental tasks, evaluation of the participants’ responses and psychological model building, were inspired by the semantics of the material conditional. Recent empirical work on indicative conditionals focuses on uncertainty. Consequently, the normative standard of reference has changed. I argue why neither logic nor standard probability (...) theory provide appropriate rationality norms for uncertain conditionals. I advocate coherence based probability logic as an appropriate framework for investigating uncertain conditionals. Detailed proofs of the probabilistic non-informativeness of a paradox of the material conditional illustrate the approach from a formal point of view. I survey selected data on human reasoning about uncertain conditionals which additionally support the plausibility of the approach from an empirical point of view. (shrink)
The de Morgan laws characterize how negation, conjunction, and disjunction interact with each other. They are fundamental in any semantics that bases itself on the propositionalcalculus/Boolean algebra. This paper is primarily concerned with the second law. In English, its validity is easy to demonstrate using linguistic examples. Consider the following: (3) Why is it so cold in here? We didn’t close the door or the window. The second sentence is ambiguous. It may mean that I suppose we (...) did not close the door or did not close the window, but I am not sure which. This `I am not sure which’ reading is irrelevant to us because it has disjunction scoping over negation. But the sentence may equally well mean (and indeed this is the preferred reading) that we didn’t close the door and did not close the window. This `neither’ reading bears out de Morgan law (2). (shrink)
This note aims at critically assessing a little-noticed proposal made by Popper in the second edition of "Objective Knowledge" to the effect that verisimilitude of scientific theories should be made relative to the problems they deal with. Using a simple propositionalcalculus formalism, it is shown that the "relativized" definition fails for the very same reason why Popper's original concept of verisimilitude collapsed -- only if one of two theories is true can they be compared in terms of (...) the suggested definition of verisimilitude. (shrink)
I argue first that attention is a (maybe the) paradigmatic case of an object-directed, non-propositional intentional mental episode. In addition attention cannot be reduced to any other (propositional or non-propositional) mental episodes. Yet, second, attention is not a non-propositional mental attitude. It might appear puzzling how one could hold both of these claims. I show how to combine them, and how that combination shows how propositionality and non-propositionality can co-exist in a mental life. The crucial move (...) is one away from an atomistic, building block picture to a more holistic, structural picture. (shrink)
Reprinted in Philosophy of Religion: An Anthology, Wadsworth 2015, 6th edition, eds Michael Rea and Louis Pojman. What is propositional faith? At a first approximation, we might answer that it is the psychological attitude picked out by standard uses of the English locution “S has faith that p,” where p takes declarative sentences as instances, as in “He has faith that they’ll win”. Although correct, this answer is not nearly as informative as we might like. Many people say that (...) there is a more informative answer. They say that, at the very least, propositional faith requires propositional belief. More precisely, they say that faith that p requires belief that p or that it must be partly constituted by belief that p. This view is common enough; call it the Common View. I have two main aims in this paper: (i) to exhibit the falsity of the Common View and the paucity of reasons for it, and (ii) to sketch a more accurate and comprehensive account of what propositional faith is. (shrink)
The philosophical case for extended cognition is often made with reference to ‘extended-memory cases’ ; though, unfortunately, proponents of the hypothesis of extended cognition as well as their adversaries have failed to appreciate the kinds of epistemological problems extended-memory cases pose for mainstream thinking in the epistemology of memory. It is time to give these problems a closer look. Our plan is as follows: in §1, we argue that an epistemological theory remains compatible with HEC only if its epistemic assessments (...) do not violate what we call ‘the epistemic parity principle’. In §2, we show how the constraint of respecting the epistemic parity principle stands in what appears to be a prima facie intractable tension with mainstream thinking about cases of propositional memory. We then outline and evaluate in §3 several lines of response. (shrink)
Most contemporary philosophical discussions of intentionality start and end with a treatment of the propositional attitudes. In fact, many theorists hold that all attitudes are propositional attitudes. Our folk-psychological ascriptions suggest, however, that there are non-propositional attitudes: I like Sally, my brother fears snakes, everyone loves my grandmother, and Rush Limbaugh hates Obama. I argue that things are as they appear: there are non-propositional attitudes. More specifically, I argue that there are attitudes that relate individuals to (...) non-propositional objects and do so not in virtue of relating them to propositions. I reach this conclusion by not only showing that attempted analyses of apparently non-propositional attitudes in terms of the propositional fail, but that some non-propositional attitudes don’t even supervene on propositional attitudes. If this is correct, then the common discussions of intentionality that address only propositional attitudes are incomplete and those who hold that all intentional states are propositional are mistaken. (shrink)
Propositionalism is the view that intentional attitudes, such as belief, are relations to propositions. Propositionalists argue that propositionalism follows from the intuitive validity of certain kinds of inferences involving attitude reports. Jubien (2001) argues powerfully against propositions and sketches some interesting positive proposals, based on Russell’s multiple relation theory of judgment, about how to accommodate “propositional phenomena” without appeal to propositions. This paper argues that none of Jubien’s proposals succeeds in accommodating an important range of propositional phenomena, such (...) as the aforementioned validity of attitude-report inferences. It then shows that the notion of a predication act-type, which remains importantly Russellian in spirit, is sufficient to explain the range of propositional phenomena in question, in particular the validity of attitude-report inferences. The paper concludes with a discussion of whether predication act-types are really just propositions by another name. (shrink)
In the article I deal with some paradoxes and errors caused by improper usage of logical and philosophical terms appearing in the arguments for existence of god and other philosophical issues. I point at rst some paradoxes coming om improper usage of propositionalcalculus as an instrument for analysis of a natural language. this language is actually not using simple sentences but rather propositional functions, their logical connections, and some replacements for variables in them. We still have (...) to deal with so called paradox of material implication. the second paragraph provides formal and metatheoretical critics of Charles Sanders Peirce’s theory of deduction, induction and abduction. I argue that what Peirce and his followers call abduction is actually deduction or some reasoning unable to describe in terms of the logic used by them. Both syllogistic and inferential theory of abduction generate some paradoxes and contradictions. In the last paragraph also some paradoxes and contradictions resulting om the theory of causation by Jan Łukasiewicz are presented. the central issue of the article is erroneous usage of the implication: in logical paraphrases of a natural language, in description of the scienti c reasoning, and in description of causality. However, my objective is not to solve all problems mentioned above but rather to open a discussion over them. (shrink)
This work studies some problems connected to the role of negation in logic, treating the positive fragments of propositionalcalculus in order to deal with two main questions: the proof of the completeness theorems in systems lacking negation, and the puzzle raised by positive paradoxes like the well-known argument of Haskel Curry. We study the constructive com- pleteness method proposed by Leon Henkin for classical fragments endowed with implication, and advance some reasons explaining what makes difficult to extend (...) this constructive method to non-classical fragments equipped with weaker implications (that avoid Curry's objection). This is the case, for example, of Jan Lukasiewicz's n-valued logics and Wilhelm Ackermann's logic of restricted implication. Besides such problems, both Henkin's method and the triviality phenomenon enable us to propose a new positive tableau proof system which uses only positive meta-linguistic resources, and to mo- tivate a new discussion concerning the role of negation in logic proposing the concept of paratriviality. In this way, some relations between positive reasoning and infinity, the possibilities to obtain a ¯first-order positive logic as well as the philosophical connection between truth and meaning are dis- cussed from a conceptual point of view. (shrink)
In recent years, the e ffort to formalize erotetic inferences (i.e., inferences to and from questions) has become a central concern for those working in erotetic logic. However, few have sought to formulate a proof theory for these inferences. To fill this lacuna, we construct a calculus for (classes of) sequents that are sound and complete for two species of erotetic inferences studied by Inferential Erotetic Logic (IEL): erotetic evocation and regular erotetic implication. While an attempt has been made (...) to axiomatize the former in a sequent system, there is currently no proof theory for the latter. Moreover, the extant axiomatization of erotetic evocation fails to capture its defeasible character and provides no rules for introducing or eliminating question-forming operators. In contrast, our calculus encodes defeasibility conditions on sequents and provides rules governing the introduction and elimination of erotetic formulas. We demonstrate that an elimination theorem holds for a version of the cut rule that applies to both declarative and erotetic formulas and that the rules for the axiomatic account of question evocation in IEL are admissible in our system. (shrink)
Vector models of language are based on the contextual aspects of language, the distributions of words and how they co-occur in text. Truth conditional models focus on the logical aspects of language, compositional properties of words and how they compose to form sentences. In the truth conditional approach, the denotation of a sentence determines its truth conditions, which can be taken to be a truth value, a set of possible worlds, a context change potential, or similar. In the vector models, (...) the degree of co-occurrence of words in context determines how similar the meanings of words are. In this paper, we put these two models together and develop a vector semantics for language based on the simply typed lambda calculus models of natural language. We provide two types of vector semantics: a static one that uses techniques familiar from the truth conditional tradition and a dynamic one based on a form of dynamic interpretation inspired by Heim’s context change potentials. We show how the dynamic model can be applied to entailment between a corpus and a sentence and provide examples. (shrink)
To understand what non-propositional content is and whether there are any such contents, we first need to know what propositional content is. That issue will be the focus of the first section of this essay. In the second section, with an understanding of propositional content in hand, we will consider representations that fail to have propositional content. In contrast to recent literature, it will be argued that metaphysical considerations concerning what’s represented, rather than linguistic considerations, are (...) a more promising way of establishing non-propositional contents. To keep the discussion containable, focus will be on representational mental states, though many of the considerations can be extended to other forms of representation. (shrink)
The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.
How can the propositional attitudes of several individuals be aggregated into overall collective propositional attitudes? Although there are large bodies of work on the aggregation of various special kinds of propositional attitudes, such as preferences, judgments, probabilities and utilities, the aggregation of propositional attitudes is seldom studied in full generality. In this paper, we seek to contribute to filling this gap in the literature. We sketch the ingredients of a general theory of propositional attitude aggregation (...) and prove two new theorems. Our first theorem simultaneously characterizes some prominent aggregation rules in the cases of probability, judgment and preference aggregation, including linear opinion pooling and Arrovian dictatorships. Our second theorem abstracts even further from the specific kinds of attitudes in question and describes the properties of a large class of aggregation rules applicable to a variety of belief-like attitudes. Our approach integrates some previously disconnected areas of investigation. (shrink)
Intentionality, or the power of minds to be about, to represent, or to stand for things, remains central in the philosophy of mind. But the study of intentionality in the analytic tradition has been dominated by discussions of propositional attitudes such as belief, desire, and visual perception. There are, however, intentional states that aren't obviously propositional attitudes. For example, Indiana Jones fears snakes, Antony loves Cleopatra, and Jane hates the monster under her bed. The present paper explores such (...) mental states in an introductory but opinionated way. (shrink)
Literature in epistemology tends to suppose that there are three main types of understanding – propositional, atomistic, and objectual. By showing that all apparent instances of propositional understanding can be more plausibly explained as featuring one of several other epistemic states, this paper argues that talk of propositional understanding is unhelpful and misleading. The upshot is that epistemologists can do without the notion of propositional understanding.
I extend the Higher-Order View of Undermining Defeat (HOVUD) defended in Melis (2014) to account for the defeat of propositional justification. In doing so, I clarify the important notion of higher-order commitment, and I make some considerations concerning the defeat of externalist epistemic warrants.
Building on the work of Peter Hinst and Geo Siegwart, we develop a pragmatised natural deduction calculus, i.e. a natural deduction calculus that incorporates illocutionary operators at the formal level, and prove its adequacy. In contrast to other linear calculi of natural deduction, derivations in this calculus are sequences of object-language sentences which do not require graphical or other means of commentary in order to keep track of assumptions or to indicate subproofs. (Translation of our German paper (...) "Ein Redehandlungskalkül. Ein pragmatisierter Kalkül des natürlichen Schließens nebst Metatheorie"; online available at http://philpapers.org/rec/CORERE.). (shrink)
A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra–Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily (...) extended to other modal logics. (shrink)
On a traditional or default view of the grasping or understanding of a singular proposition by an individual, it is assumed to be a unitary or holistic activity. However, naturalistic views of cognition plausibly could analyze propositional thinking in terms of more than one distinctive functional stage of cognitive processing, suggesting at least the potential legitimacy of a non-unitary analysis of propositional grasping. We outline a novel dual-component view of this kind, and show that it is well supported (...) by current cognitive science research. (shrink)
In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. (...) Firstly, it is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up. (shrink)
What the authors attempt to address in this paper is a Kantian question: not whether, but how is cross -cultural understanding possible? And specifically, what is a more effective approach for cross -cultural understanding? The answer lies in an analysis of two different models of cross -cultural understanding, that is, propositional and hermeneutic understanding. To begin with, the author presents a linguistic interpretation of culture, i.e., a culture as a linguistically formulated and transmitted symbolic system with its conceptual core (...) as a scheme of basic cultural presuppositions, which it referred to as a cultural language. After exploring the essential role of cultural presuppositions in cross -cultural understanding, the author discusses the traditional model of cross -cultural understanding, namely, the propositional model. Through critically examining the two popular versions of the propositional model, i.e., the projective approach and the adoptive approach to cross -cultural understanding, it is found that cross -cultural propositional understanding is doomed to failure. To move us beyond the absolutism -relativism trap embedded within propositional understanding, the author first introduces and discusses Hans-Georg Gadamer’s hermeneutic understanding, and then applies Hans-Georg Gadamer’s hermeneutic model of understanding to cross -cultural understanding. It is finally concluded that cross -cultural understanding is essentially hermeneutic—including the case of cultural learning, not propositional. Therefore, cross -cultural understanding is hermeneutically possible. (shrink)
John Venn has the “uneasy suspicion” that the stagnation in mathematical logic between J. H. Lambert and George Boole was due to Kant’s “disastrous effect on logical method,” namely the “strictest preservation [of logic] from mathematical encroachment.” Kant’s actual position is more nuanced, however. In this chapter, I tease out the nuances by examining his use of Leonhard Euler’s circles and comparing it with Euler’s own use. I do so in light of the developments in logical calculus from G. (...) W. Leibniz to Lambert and Gottfried Ploucquet. While Kant is evidently open to using mathematical tools in logic, his main concern is to clarify what mathematical tools can be used to achieve. For without such clarification, all efforts at introducing mathematical tools into logic would be blind if not complete waste of time. In the end, Kant would stress, the means provided by formal logic at best help us to express and order what we already know in some sense. No matter how much mathematical notations may enhance the precision of this function of formal logic, it does not change the fact that no truths can, strictly speaking, be revealed or established by means of those notations. (shrink)
In this paper we focus our attention on tableau methods for propositional interval temporal logics. These logics provide a natural framework for representing and reasoning about temporal properties in several areas of computer science. However, while various tableau methods have been developed for linear and branching time point-based temporal logics, not much work has been done on tableau methods for interval-based ones. We develop a general tableau method for Venema's \cdt\ logic interpreted over partial orders (\nsbcdt\ for short). It (...) combines features of the classical tableau method for first-order logic with those of explicit tableau methods for modal logics with constraint label management, and it can be easily tailored to most propositional interval temporal logics proposed in the literature. We prove its soundness and completeness, and we show how it has been implemented. (shrink)
The folk Psychology frames propositional attitudes as fundamental theoretical entities for the construction of a model designed to predict the behavior of a subject. A trivial, such as grasping a pen and writing reveals - something complex - about the behavior. When I take a pen and start writing I do, trivially, because I believe that a certain object in front of me is a pen and who performs a specific function that is, in fact, that of writing. When (...) I believe that the object that stands before me is a pen, I am in relation to "believe" with the propositional content: that in front of me is a pen. Philosophers of the proposition, from Frege onwards, have dedicated their studies to the analysis of what kinds of entities are the propositional attitudes. Jerry Fodor says that now, the proper prediction of the psychology of common sense, can not be questioned and that the propositional attitudes represent the most effective way to describe our behavior. What Fodor says, however, is that propositional attitudes function, but not how they work. Most philosophers interested in the issue, we are dedicated to the search for a theory that can account consistently both a semantics for propositional attitudes, both of these entities that seem to cause the behavior of a rational subject. There are two main paradigms in the theory of the proposition that contributed to the discussion of the propositional attitudes. One is the one that begins with Gottlob Frege, the other with Bertrand Russell. Defenders of Frege argue that the paradigm scrub objects and properties can not be constituents of the propositional content which have a purely conceptual. In other words, the philosophers belonging to the paradigm of Frege, but not all, mean that you can test in a rigorous way the truth conditions of propositional attitudes. Who defends the russellian’s paradigm argues that the propositional content are made by the objects and properties on which propositional attitudes relate. The purpose of this article is not to rebuild - in detail - both paradigms, nor to reconstruct one but, in a sense, my work will be a completely partial objective is to demonstrate how the paradigm is more profitable russell not only to make a coherent semantic theory for propositional attitudes2 but also to predict the behavior of a rational subject thing, completely innovative, given the repeated objections in contemporary literature3. At the end of this paper will be drafted a proposal to build a consistent model to predict the behavior,of a rational agent, based on a referential theory of propositional attitudes. (shrink)
To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian (...) infinitesimal calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)
The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. (...) Next, the geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)
It is shown that Gqp↑, the quantified propositional Gödel logic based on the truth-value set V↑ = {1 - 1/n : n≥1}∪{1}, is decidable. This result is obtained by reduction to Büchi's theory S1S. An alternative proof based on elimination of quantifiers is also given, which yields both an axiomatization and a characterization of Gqp↑ as the intersection of all finite-valued quantified propositional Gödel logics.
I explore the possibility that propositional attitudes are not basic in folk psychology, and that what we really ascribe to people are relations to individuals, those that the apparently propositional contents of beliefs, desires, and other states concern. In particular, the relation between a state and the individuals that it tracks shows how ascription of propositional attitudes could grow out of ascription of relations between people and objects.
Propositional attitude sentences, such as John believes that snow is white, are traditionally taken to express the holding of a relation between a subject and what ‘that’-clauses like ‘that snow is white’ denote, i.e. propositions. On the traditional account, propositions are abstract, mind- and language-independent entities. Recently, some have raised some serious worries for the traditional account and thought that we were mistaken about the kind of entities propositions are. Over the last ten years there has then been a (...) boom of accounts of propositions in terms of mental acts . But Friederike Moltmann has recently suggested that in accounting for attitudes we should forget about mind- and language-independent entities and acts and follow Twardowski in focusing instead on attitudinal objects, which are the products of our mental life. In this paper, I will focus on some semantic problems that any product-based account seems to face. Moreover, I will show that product-based accounts may be also criticised on ontological grounds. My conclusion will be that we lack a reason to think that in accounting for propositional attitudes we should focus on the alleged products of our mental lives. (shrink)
This paper is the twin of (Duží and Jespersen, in submission), which provides a logical rule for transparent quantification into hyperprop- ositional contexts de dicto, as in: Mary believes that the Evening Star is a planet; therefore, there is a concept c such that Mary be- lieves that what c conceptualizes is a planet. Here we provide two logical rules for transparent quantification into hyperpropositional contexts de re. (As a by-product, we also offer rules for possible- world propositional contexts.) (...) One rule validates this inference: Mary believes of the Evening Star that it is a planet; therefore, there is an x such that Mary believes of x that it is a planet. The other rule validates this inference: the Evening Star is such that it is believed by Mary to be a planet; therefore, there is an x such that x is believed by Mary to be a planet. Issues unique to the de re variant include partiality and existential presupposition, sub- stitutivity of co-referential (as opposed to co-denoting or synony- mous) terms, anaphora, and active vs. passive voice. The validity of quantifying-in presupposes an extensional logic of hyperinten- sions preserving transparency and compositionality in hyperinten- sional contexts. This requires raising the bar for what qualifies as co-denotation or equivalence in extensional contexts. Our logic is Tichý’s Transparent Intensional Logic. The syntax of TIL is the typed lambda calculus; its highly expressive semantics is based on a procedural redefinition of, inter alia, functional abstraction and application. The two non-standard features we need are a hyper- intension (called Trivialization) that presents other hyperintensions and a four-place substitution function (called Sub) defined over hy- perintensions. (shrink)
Motivated by H. Curry’s well-known objection and by a proposal of L. Henkin, this article introduces the positive tableaux, a form of tableau calculus without refutation based upon the idea of implicational triviality. The completeness of the method is proven, which establishes a new decision procedure for the (classical) positive propositional logic. We also introduce the concept of paratriviality in order to contribute to the question of paradoxes and limitations imposed by the behavior of classical implication.
This paper critiques the view, widely held by philosophers of mind and cognitive scientists, that psychological explanation is a matter of ascribing propositional attitudes (such as beliefs and desires) towards language-like propositions in the mind, and that cognitive mental states consist in intentional attitudes towards propositions of a linguistic quasi-linguistic nature. On this view, thought is structured very much like a language. Denial that propositional attitude psychology is an adequate account of mind is therefore, on this view, is (...) tantamount to eliminative materialism, the denial that human beings are thinking beings. -/- I dispute this on the basis of recent work in cognitive psychology and artificial intelligence. Mental models theory, on which thought is better understood as nonpropositional intentional psychology, accords better with the evidence and offers an alternative view to propositional attitude psychology -- one that means that the denial that that is propositional is not eliminative. However, I argue that propositional attitude psychology is a useful idealization, as classical mechanics is of relativity theory, strictly but not radically false, and useful for prediction and indeed, as long as its idealized character is born in mind, for explanation of behavior. (shrink)
This article presents modal versions of resource-conscious logics. We concentrate on extensions of variants of linear logic with one minimal non-normal modality. In earlier work, where we investigated agency in multi-agent systems, we have shown that the results scale up to logics with multiple non-minimal modalities. Here, we start with the language of propositional intuitionistic linear logic without the additive disjunction, to which we add a modality. We provide an interpretation of this language on a class of Kripke resource (...) models extended with a neighbourhood function: modal Kripke resource models. We propose a Hilbert-style axiomatisation and a Gentzen-style sequent calculus. We show that the proof theories are sound and complete with respect to the class of modal Kripke resource models. We show that the sequent calculus admits cut elimination and that proof-search is in PSPACE. We then show how to extend the results when non-commutative connectives are added to the language. Finally, we put the l.. (shrink)
This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how phenomenal consciousness and gradational possible-worlds models in Bayesian perceptual psychology relate to epistemic modal space. The book demonstrates, then, how epistemic modality relates to the computational theory of mind; metaphysical modality; deontic modality; logical modality; the types of mathematical modality; to the (...) epistemic status of undecidable propositions and abstraction principles in the philosophy of mathematics; to the apriori-aposteriori distinction; to the modal profile of rational propositional intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Examining the nature of epistemic logic itself, I develop a novel approach to conditions of self-knowledge in the setting of the modal μ-calculus, as well as novel epistemicist solutions to Curry's, the liar, and the knowability paradoxes. Solutions to the Julius Caesar Problem, and to previously intransigent issues concerning the first-person concept, the distinction between fundamental and derivative truths, and the unity of intention and its role in decision theory, are developed along the way. (shrink)
This dissertation concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The dissertation demonstrates how phenomenal consciousness and gradational possible-worlds models in Bayesian perceptual psychology relate to epistemic modal space. The dissertation demonstrates, then, how epistemic modality relates to the computational theory of mind; metaphysical modality; deontic modality; logical modality; the types of mathematical modality; to the (...) epistemic status of undecidable propositions and abstraction principles in the philosophy of mathematics; to the apriori-aposteriori distinction; to the modal profile of rational propositional intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Examining the nature of epistemic logic itself, I develop a novel approach to conditions of self-knowledge in the setting of the modal μ-calculus, as well as novel epistemicist solutions to Curry's, the liar, and the knowability paradoxes. Solutions to previously intransigent issues concerning the first-person concept; the distinction between fundamental and derivative truths; and the unity of intention and its role in decision theory, are developed along the way. (shrink)
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